
where \(\gamma \)  is the fraction factor that accounts for how much of the horizon mass turns into the primordial blackhole, \(M_H=4\pi M_{\rm P}^2/H\)  is the horizon mass, and the subscript \({\rm f}\)  as in \(H_{\rm f}\)  indicates a quantity at the formation time. We shall use \(\gamma = 0.2\)  as suggested by simple analytical estimations [1]}.
From the entropy conservation, we find that the Hubble rate at the formation time, \(H_{\rm f}\) , is related to the Hubble rate at present, \(H_0\) , as
\(H_{\rm f} = H_0 \frac{\Omega _{\rm rad,0}^{1/2}}{a_{\rm f}^2}\left(\frac{g_{*,0}}{g_{*,{\rm f}}}\right)^{\frac{1}{6}}\,,\) 
